
Firstly, for a given angular acceleration profile, the non-holonomic motion model is affine with respect to linear velocity. Further, fixing angular acceleration also ensures that the posture profile of the vehicle does not change in the velocity optimization layer. Since the Minkowski sum only depends on the relative posture of the vehicle and obstacle and not on their relative distance, the velocity optimization has the same DC structure which has been reported for linear systems with circular vehicles/obstacles [1]}, [2]}, [3]}). We are not aware of any work which has highlighted such structure in non-holonomic trajectory optimization with convex polygon shapes.

We empirically show through extensive simulations that the proposed alternating optimization can afford larger trust regions as compared to joint optimizations which simultaneously optimizes in the space of angular acceleration and linear velocity. Consequently, the alternating optimization provides some gain in computation time over the joint formulations.

The proposed circle approximation using the concept of Minkowski sum and minimum bounding circle leads to significantly less conservative approximation than representing each polygon individually through a circumscribing circle. Further, this is also an improvement over the approach presented in [4]} where polygons are represented as multiple overlapping circles. In particular, as compared to these cited works, the proposed approach leads to reduced number of constraints in the trajectory optimization. For example, if three overlapping circles are used to represent just the vehicle polygon, then the trajectory optimization in [4]} would have three times more number of constraints than the proposed approach.

